500 Mr. Ivory on the Theory of the Astronomical Refractions, 



Continuing to integrate in like manner, we shall find after 

 n successive operations, 



which is obviously a positive quantity. 

 By expanding, we get 



and, by performing the differential operations, 



1 rf"*" , , t t n — \\ 



+ 2.^^&c. 

 Now, because t = — , if we put, 



^''' ~ 1.2.3...W * rZ/" ' 



we shall have 



... , n+\ X w-1 w+l.w+2 ^2 



^ '^ 7M 1 2 mr 1.2 



Another form may be given to this function ; for, without any 

 variation in quantity, ^ and 1 — / may be interchanged, not 

 only in 



*« = tn{\_t)% 



but in all its differentials, observing that the results equal in 

 quantity will have opposite signs when the number of differ- 

 entiations is odd, and the same sign when the number is even. 



Now if, instead of ^ = — » we substitute 1— ^ = 



m m 



shall have 



we 



* 



/ \ I fi '*+! m—x 



n — l n-{-l.7i + 2 (m — xY _ 

 2 nr 1 .2 



